Inversion of the Weyl integral transform and the radon transform on Rn using generalized wavelets

We invert the Weyl integral transform by means of a generalized continuous wavelet transform on the half line associated with the Bessel operatorL , >–1/2. Next, we use the connection between radial classical wavelets onR ⁿ and generalized wavelets associated with the Bessel operatorL( _n–2)/2 to derive new inversion formulas for the Radon transform onR ⁿ ,n2.     

Die Laguerre-Pinney-Transformation

Ohne Zusammenfassung

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The Radon transform on hyperbolic space

The Radon transform that integrates a function in ⁿ , the n-dimensional hyperbolic space, over totally geodesic submanifolds with codimension 1 and the dual Radon transform are investigated in this paper. We prove inversion formulas and an inclusion theorem for the range.     

Integral geometry and three-dimensional reconstruction of randomly oriented identical particles from their electron microphotos

A new method for the three-dimensional reconstruction of a structure from projections of randomly oriented particles on a plane is proposed. Reconstruction is performed in two steps. First, we find mutual orientation of particles, i.e., Euler angles, describing the angle of one particle with respect to another. Almost all the paper is devoted to solving this problem. Then we perform the three-dimensional reconstruction of an object from its projections in already-known directions.The stability of the method with respect to experimental errors is shown. Three-dimensional reconstruction of asymmetric biological objects might be one of its applications.     

Methods of integral geometry and recovering a function with compact support from its projections in unknown directions

A method for recovering a function with compact support on a plane from its Radon transform is suggested. In particular, from the given projections of a function in unknown directions, these directions may be recovered (up to a rotation or a reflection), if there are 7 of them. This result may be applied in tomography.     

About asymptotic formulas for the inverse Radon transform

We give a proof and generalizations of the Gelfand-Graev asymptotic formula (formulated in 1962 for the odd-dimensional inverse Radon transform).     

On the character of operational solutions of the time-fractional diffusion equation

The nonhomogeneous time-fractional diffusion equation is analyzed in the frames of the Mikusiński calculus. The exact and the approximate solution of the considered problem are constructed and their character is analyzed.     

Series like Taylor's series

A theorem and some examples are given concerning the convergence, in a space of generalized functions, of power series whose terms contain successive derivatives of a given function. One example is the Euler-Maclaurin sum formula, where there are some novelties.     

Finding exact solutions of nonlinear PDEs using the natural decomposition method

In this work, we implement the natural decomposition method (NDM) to solve nonlinear partial differential equations. We apply the NDM to obtain exact solutions for three applications of nonlinear partial differential equations. The new method is a combination of the natural transform method and the Adomian decomposition method. We prove some of the properties that are related to the natural transform method. The results are compared with existing solutions obtained by other methods, and one can conclude that the NDM is easy to use and efficient. Copyright © 2016 John Wiley & Sons, Ltd.     

On extending backwards positive definite sequences

The two-sided Hamburger moment problem¹, also called the strong one [4], has been extensively studied in recent years in connection with rational approximation. We propose to consider the question of when a sequence, say {a _n } _n=0 can be extended backwards so that the resulting sequence {a _n } _n=–N has an integral representation of the Hamburger type. This was settled (without any proof) under different circumstances in [6]. Here we wish to discuss this completely, as well as the possibility of extending {a _n } _n=0 to {a _n } _n– .     

Transformations of matricial α-Stieltjes non-negative definite sequences

We investigate the inner structure of power moment sequences of matrix measures on the right semiaxis [α,+∞)$[\alpha ,+\infty )$, where α is a given real number. To a given matrix sequence, we associate in a bijective way a new sequence of matrices, which we call the right α  -Stieltjes parametrization. Thereby, one-to-one correspondences between power moment sequences on the right semiaxis [α,+∞)$[\alpha ,+\infty )$ with additional properties and particular sequences of non-negative Hermitian matrices are established. We consider distinguished transformations of matrix sequences the study of which was suggested by considering some natural transformations of matrix measures on an interval. A main theme is to describe the right α-Stieltjes parametrization of the transformed sequence in terms of the right α-Stieltjes parametrization of the original sequence.     

The boundedness of the Cauchy singular integral operator in weighted Besov type spaces with uniform norms

The mapping properties of the Cauchy singular integral operator with constant coefficients are studied in couples of spaces equipped with weighted uniform norms. Recently weighted Besov type spaces got more and more interest in approximation theory and, in particular, in the numerical analysis of polynomial approximation methods for Cauchy singular integral equations on an interval. In a scale of pairs of weighted Besov spaces the authors state the boundedness and the invertibility of the Cauchy singular integral operator. Such result was not expected for a long time and it will affect further investigations essentially. The technique of the paper is based on properties of the de la Vallée Poussin operator constructed with respect to some Jacobi polynomials.     

Hypoelliptic Jacobi–Dunkl Convolution of Distributions

In this paper we give a necessary and sufficient condition in terms of the Jacobi–Dunkl transform in order that a Jacobi–Dunkl convolution of distributions is hypoelliptic.     

Additive endomorphisms of rings

We characterize finite rings all of whose additive endomorphisms are multiplicative also.This work was supported by Alabama State University Faculty Research Grant R-78-6.     

Generalizations of the Busemann-Petty problem for sections of convex bodies

We present generalizations of the Busemann-Petty problem for dual volumes of intermediate central sections of symmetric convex bodies. It is proved that the answer is negative when the dimension of the sections is greater than or equal to 4. For two- three-dimensional sections, both negative and positive answers are given depending on the orders of dual volumes involved, and certain cases remain open. For bodies of revolution, a complete solution is obtained in all dimensions.